Introduction

This final chapter is devoted to establishing sufficient conditions for two OU measures μ and ν on a sigma compact manifold X to be ‘homeomorphic’. Recall that this means there exists a homeomorphism h of X onto itself satisfying ν (A) = μ (h(A)) for all measurable sets A, which we write simply as ν = μh. The general result of this type is due to Berlanga and Epstein [38], who use in their proof the important special case of the theorem for compact manifolds due to Oxtoby and Ulam [88] and von Neumann [105]. In Chapter 9 we developed various corollaries of the compact version of the theorem, which was stated for the unit cube in Theorem 9.1. The version for a general compact manifold is given here as Corollary A2.6, and the version for sigma compact manifolds is given here as Theorem A2.8. In this chapter we outline the original proofs of the compact case and then the sigma compact case. As such, this is the only chapter of the book that is not based on the work of the authors.

Recall that a Borel measure on a manifold is called an OU measure if it is zero for points, positive on open sets, and zero on the boundary set. Since singleton sets, open sets, and the boundary set are all preserved under homeomorphism, it follows that any Borel measure homeomorphic to an OU measure must also be an OU measure.